This paper studies effective scenarios in Distributionally Robust Optimization (DRO) problems defined on a finite number of realizations (also called scenarios) of the uncertain parameters. Effective scenarios are critical scenarios in DRO in the sense that their removal from the support of the considered distributions alters the optimal value. Ineffective scenarios are those whose removal do not alter the optimal value. In this paper, we first link the effectiveness of a scenario to its worst-case probability being always positive or uniquely zero under a general distance-based ambiguity set. We then narrow our focus to DROs with ambiguity sets formed via the Wasserstein distance (denoted DRO-W), and we provide easy-to-check sufficient conditions to identify the effectiveness of scenarios for this class of problems. When the Wasserstein distance is equivalent to the total variation distance (i.e., when the transportation cost between scenarios is zero if they are the same and one if they are different), the easy-to-check conditions for DRO-W presented in this paper recover the ones presented in the literature for DRO formed via the total variation distance as a special case. The numerical findings highlight the relationship between scenario effectiveness and the attributes of the transportation cost between scenarios that constitute the Wasserstein distance, revealing useful insights.